Time-dependent scaling relations and a cascade model of turbulence

Abstract

We study the time-dependent solutions of a nonlinear cascade model for homogeneous isotropic turbulence first introduced by Novikov & Desnyansky. The dynamical variables of the model are the turbulent kinetic energies in discrete wave-number shells of thickness one octave. The model equations contain a parameter C whose size governs the amount of energy cascaded to small wavenumbers relative to the amount cascaded to large wavenumbers. We show that the equations permit scale-similar evolution of the energy spectrum. For 0 ≤ C ≤ 1 and no external force, the freely evolving energy spectrum displays the Kolmogorov k power law, and the total energy decreases in time as a power t^−w, where the exponent w depends on the value of C. Grid-turbulence experiments seem to favour a value of C in the range 0·3-0·6. In the presence of an external stirring force acting near a wavenumber k₀, the model predicts, in addition to the Kolmogorov k spectrum for k > k₀, a scale-similar flow of energy to wavenumbers k < k₀. This backward energy flow falls off as a power law in time, and establishes a stationary energy spectrum for k < k₀ which is a power law in k less steep than k. We discuss the similarity of the behaviour of the model for C > 1 to the behaviour of turbulent fluid for a spatial dimensionality near 2. The model is shown to approach the Kovasznay and the Leith diffusion approximation equations in the limit in which the thickness of the wavenumber shells approaches zero. However, the cascade model with finite shell thicknesses appears to behave in a more physically reasonable way than the limiting differential equations.